The work described below has been used in the context of a larger joint project, Using Conway's Cosmological Sequence for Pseudo-Random Number Generation: pdf ps.
What is the next number?
One often tries to take an arithmetic approach to finding the next number in the sequence. That method however will not work. To find the next number, one must successively tallying the number of consecutive digits in a run and appending this and the digit to the new string. So looking at the last round of our sequence (111221), we can see that there are 3 ones in a row, 2 twos in a row, and 1 one. The next number in the sequence will be 312211. This sequence is known as John Conway's Cosmological sequence and I have devised a method to use this sequence to generate a Pseudo-Random number for use as a One-Time Pad.
The reverse method repeats rather quickly as well as the reverse and replace method. The middle-permute method repeats after about 3 million, which is ok but still not good enough to use. Ideally we would want a sequence that repeats after about 10 billion.
Although current day pseudo-random number generators do a good job of spawning random numbers, the question of finding a better generator always remains eternal. Our methodology of using John Conway's Cosmological sequence is completely atypical in the field of generating random numbers. However, we believe that such a method has promise and, depending on certain parameters, may even prove better than the norm.
W. Valencia and P. Del Mundo "Using Conway's Cosmological Sequence for Pseudo-Random Number Generation" UMCP College Park 2003.